Accrual Accounting, Informational Sufficiency, and Equity Valuation

Transcription

1 Accrual Accounting, Informational Sufficiency, and Equity Valuation Alexander A. Nezlobin Graduate School of Business Stanford University Abstract This paper studies accrual accounting and equity valuation in the context of a firm that makes repeated and overlapping investments in productive capacity. The analysis identifies a particular accrual accounting (depreciation) rule which is termed replacement cost accounting because the book value of existing capacity assets is set equal to the value that such assets would have if a competitive market were to exist for used assets. It is shown that replacement cost accounting aggregates past investment decisions of the firm without a loss of value-relevant information. In particular, the intrinsic value of the firm can then be expressed as a function of current accounting data and projections of growth in the firm s output market. Further, it is shown that replacement cost accounting is essentially the only accounting rule with this informational sufficiency property. In many environments of interest, standard depreciation rules, such as the straight-line rule, will not coincide with replacement cost accounting. Nonetheless, even informationally insufficient accounting rules are shown to be useful to investors. In particular, the analysis in this paper characterizes a best value estimator and derives upper bounds on the valuation errors associated with alternative accounting rules. Ph.D. Candidate in Accounting, nezlobin@stanford.edu. I would like to thank my academic advisor, Stefan Reichelstein, for his continuous support, guidance, and encouragement. I am also grateful to William Beaver, Anne Beyer, Ilan Guttman, Maureen McNichols, Joseph Piotroski, Madhav Rajan, and Daniel Taylor for their comments and suggestions. Last but not least, I wish to acknowledge the advice and patient support of Nikolai Nezlobin and Polina Nezlobin. 1

2 1 Introduction This paper examines the informativeness of alternative accrual accounting rules for the purpose of firm valuation. I model a firm that undertakes a sequence of investment decisions in productive capacity. The firm s accounting system aggregates the resulting investment history into current financial statements. It is shown that for certain accounting rules this aggregation process does not entail a loss of value-relevant information, and investors will be able to assess the firm s value correctly by observing only the aggregate data. Such rules will be referred to as informationally sufficient. For informationally sufficient rules, I derive valuation equations that express the firm s intrinsic value in terms of the accounting numbers observed by investors. In contrast, for informationally insufficient accounting rules, it is generally impossible to solve the valuation problem precisely. Yet, if investors have beliefs about the investment history of the firm, then they can use the information in financial statements to update these beliefs. For informationally insufficient rules, I characterize the value estimator that uses only the observed data as inputs and minimizes the mean squared error. It is shown that the error of this estimator is bounded by a measure of distance between the accounting rule in place and the closest informationally sufficient rule. Earlier theoretical literature on accounting based valuation has been largely silent on the relative advantages of alternative accrual accounting rules in providing information useful to investors. I seek to address two limitations inherent in earlier valuation studies such as Ohlson (1995), Feltham and Ohlson (1995), and Ohlson and Juettner-Nauroth (2005). First, in these papers, the firm s underlying transactions, as well as the accounting rules employed, are not modeled beyond their most basic properties, like the clean surplus relation. 1 Therefore, assumptions on the behavior of the time-series of accounting numbers (e.g. linear information dynamics) are inherently joint constraints on the economic environment of the firm and the accounting rules in use. 2 My analysis seeks to disentangle the economic and reporting factors that affect the time-series of accounting numbers. Second, none of these models explicitly articulates limitations on the information available to outsiders. With- 1 This criticism applies to earlier studies in varying degrees. More elaborate models of the firm s transactions and accounting rules can be found in Ohlson and Zhang (1998), Zhang, G. (2000), and Zhang, X. (2000). On the other hand, in his discussion of the model in Ohlson and Juettner-Nauroth (2005), Penman (2005) poses the question: "Where s the accounting?" 2 Several empirical studies challenged the linear information dynamics assumption of Ohlson (1995). See, for instance, Myers (1999) and Dechow, Hutton, and Sloan (1999). Discussing this issue, Kothari (2001) points out the following direction for future research: "While an autoregressive process in residual income as a parsimonious description is economically intuitive, there is nothing in economic theory to suggest that all firms residual earnings will follow an autoregressive process at all stages in their life cycle. A more fruitful empirical avenue would be to understand the determinants of the autoregressive process or deviations from that process as a function of firm, industry, macroeconomic, or international institutional characteristics." 2

3 out information asymmetry, it is difficult, if not impossible, to demonstrate informational advantages of particular accounting rules over others, including cash accounting. This paper models the activities of a firm as a sequence of capital investments in productive capacity. The firm uses its capacity to deliver goods and services which generate revenues. The accounting rules may reflect the productivity pattern of the firm s assets and provide aggregate information on the investments undertaken. Depreciation is the only accrual in my model. This focus was chosen for two reasons. First, depreciation is arguably the largest accrual in many industries. Second, earlier literature on performance measurement has shown that insights obtained in connection with capital investments and depreciation do carry over to other accrual accounting items (see, for instance, Dutta and Reichelstein, 2005). A central point of departure for my analysis is that investors must rely only on limited publicly available information for valuation purposes. Specifically, I assume that investors observe only the latest financial statements, rather than the whole investment history. Clearly, if investors could observe all the past transactions of the firm, it would be impossible to differentiate between alternative accounting rules, because there would be no need to aggregate information about the underlying transactions. The main goal of depreciation charges in my model is to aggregate information about past investments such that single period financial statements are sufficient for valuation. Depreciation schedules with this property are referred to as informationally sufficient. An additional feature of my model is the assumption that the projections of future output market growth rates cannot be incorporated into the accounting numbers. Consequently, the requirement for an informationally sufficientruleisthatitmustprovideenoughinformation to value the company under alternative assumptions about future growth opportunities. This requirement is broadly consistent with the perspective taken by standard setters in SFAC No. 1: "Financial accounting is not designed to measure directly the value of a business enterprise, but the information it provides may be helpful to those who wish to estimate its value... Although financial reporting should provide basic information to aid them [investors, creditors, and others] they do their own evaluating, estimating, predicting, assessing, confirming, changing, or rejecting." Since I assume that growth opportunities cannot be incorporated into the accounting data, fair value accounting, which would result in book values equal to market values, is rendered infeasible in the context of the present model. 3

4 My framework of a firm undertaking a sequence of overlapping capacity investments builds on Rogerson (2008a, 2008b). In this context, it is possible to identify the marginal cost of providing an additional unit of capacity for one period, holding capacity levels in other periods fixed. It can be shown that this marginal cost is equal to the rental price of a unit of capacity in a hypothetical perfectly competitive rental market. Rogerson (2008b) demonstrates that there exists a particular depreciation schedule, the Relative Replacement Cost, or RRC rule, under which the historical cost of capacity is equal to its hypothetical rental price. The first result of my paper applies this idea to the valuation problem. Given the RRC rule, it is shown that book value can be interpreted as the replacement cost of assets in place at any point in time. Therefore, I refer to the RRC rule as replacement cost accounting. Applying the residual income valuation formula, I then demonstrate that the firm s value is equal to the replacement cost of its assets in place plus the present value of its future optimized economic profits, where the measure of capacity costs in the computation of profits is the replacement cost of capacity utilized in a given period. The second result of this paper is that, under certain assumptions, replacement cost accounting is informationally sufficient, i.e., it provides enough information for a correct assessment of the firm s intrinsic value based solely on accounting data. Specifically, I show that if the output market grows proportionately at all price levels, investors can predict future economic profits if they know current economic profitsandrelyontheirprojections of growth in the firm s product market. I formally identify two-value relevant dimensions of the investment history - the replacement cost of assets in place and the replacement cost of capacity utilized in the latest period. If the difference between two investment histories is orthogonal to both value-relevant dimensions, then two firmsthathaveexperiencedthesehistorieswillhavethesamevalue. On the other hand, differences along the value-relevant dimensions can lead to differences in valuation. Under replacement cost accounting, the replacement cost of assets in place is equal to the book value and the replacement cost of capacity utilized is equal to the sum of depreciation and a cost-of-capital charge on the beginning-of-period book value. Further, I show that if the replacement cost rule is modified by some partial direct expensing of new investments, it remains informationally sufficient. I label this family of rules generalized replacement cost accounting and derive an exact valuation formula for this family of accounting rules. My third major result is that generalized replacement cost accounting is unique in its informational sufficiency. To prove this, I invoke a basic result from aggregation theory. Applications of this theory have a long tradition in accounting (e.g. Lev, 1968 and Ijiri, 4

5 1967; see also Arya et. al., 2000 for a more recent reference). My uniqueness result rests on theargumentthatinorderforfinancial statements to permit valuation for a broad range of possible output market growth projections, an informationally sufficient rule must preserve both value-relevant dimensions through book value and depreciation. This requirement implies that book value must be always proportional to the replacement cost of assets in place, i.e., the rule in use has to conform to generalized replacement cost accounting. Replacement cost accounting should be regarded as a normative benchmark rather than a description of current financial accounting practice, where straight-line depreciation is prevalent for fixed assets. For informationally insufficient rules, I also perform an analysis in the spirit of aggregation theory to bound the minimum error that investors can achieve in the valuation process. It is assumed that investors form beliefs about the firm s investment history and then update their beliefs after observing the financial statements. My finding on the uniqueness of generalized replacement cost accounting shows that informationally insufficient accounting rules are incapable of fully resolving the uncertainty related to the value-relevant dimensions of the firm s history. Therefore, the firm s intrinsic value will have some residual variance after conditioning on the accounting data. I derive an upper bound on this variance which can be interpreted as the distance between the accounting rule actually used and the closest informationally sufficient rule. In addition, my findings characterize the value estimator minimizing the mean squared error for investors. The notion of proper accrual accounting and the economic significance of "good" accounting have been explored in different strands of the accounting literature. Beaver and Dukes (1974) identify depreciation schedules for various types of assets, such that accounting rates of return equal the economic rates of return. Rajan, Reichelstein, and Soliman (2007) characterize the biases in accounting rates of return that result from applying depreciation schedules that do not match the productivity profile of assets. In the managerial literature, certain accounting rules were shown to provide goal congruent incentives for a manager to whom investment decisions have been delegated, but whose decisions are subject to potential horizon biases. 3 This paper provides a new perspective on the usefulness of alternative accrual accounting rules in the context of a forward looking equity valuation problem, absent any incentive or contracting issues. The remainder of this paper is organized as follows. The next section illustrates the concept of informational sufficiency by means of an example. In Section 3, the economic and reporting model of the firm is presented and an expression for the firm s value is derived. Section 4 provides the general concept of informational sufficiency and identifies accounting rulesthathavethissufficiency property. In contrast, the valuation properties of information- 3 See, for example, Rogerson (1997, 2008b), Reichelstein (1997, 2000). 5

6 ally insufficient accounting systems are examined in Section 5. This section also discusses empirical implications of the model. Concluding remarks are provided in the last section of the paper. 2 An Illustrative Example This section presents an example that illustrates the concept of informational sufficiency. The role of accrual accounting is to aggregate information about past transactions of the firm in a manner that preserves information essential to investors. The example shows that in this regard some accounting rules are more effective than others. In particular, I consider asimplified model in which a firm employs a single type of capital assets whose productivity is constant over their useful life. First, it will be shown that straight-line accounting is informationally insufficient for such assets, meaning that the valuation problem is not solvable if investors observe only the latest period financial statements based on straight-line depreciation. In contrast, an alternative depreciation policy, the annuity rule, aggregates past investment data without a loss of value-relevant information and enables investors to value the company based solely on current aggregate accounting data. Consider a firm which invests in capacity, produces a single type of product, and then sells this product to an outside market. In the simplified model, capacity is generated by capital assets with a useful life of four periods. Each asset is idle in the acquisition period and then provides capacity to produce one unit of output in each of the following four periods. This productivity pattern will be referred to as one-hoss shay productivity. 4 The price of capital assets is constant over time and is normalized to unity, so that an investment of I t 4 dollars in period t 4 generates I t 4 units of capacity in periods t 3, t 2, t 1, t. The total capacity available in period t is therefore given by: K t = I t 1 + I t 2 + I t 3 + I t 4. To make the valuation problem particularly simple, assume that the firm faces a kinked demand curve of the following nature in its output market. Any quantity of output up to some maximum level, q max, can be sold at a price p that is high enough to cover the costs of production. Beyond q max, the output price declines so rapidly that the firm never finds it optimal to supply more than q max. Further, assume that demand is stationary over time. 4 This term originates from the poem "The Deacon s Masterpiece, or, the Wonderful One-Hoss Shay: a Logical Story" written by Oliver W. Holmes in 1858, in which a shay is described that does not require repairs for a hundred years and then falls apart "all at once, and nothing first". The term is widely used in theeconomicliteratureonregulation. 6

7 Under these assumptions, it is optimal for the firm to generate capacity of exactly q max in every period and sell all its output at price p. Revenues in period t + τ are then equal to R t+τ = pq max = pk t+τ. The value of the firm at date t is defined as the present value of its future cash flows: V t = TX (R t+τ I t+τ ) γ τ, τ=1 where future investments are assumed to be chosen optimally, and γ = 1 is the appropriate 1+r discount factor. On the optimal path, K t+τ must be equal to q max for any τ,soineachperiod the firm will exactly replace the investment that goes offline in that period: I t+τ = I t+τ 4 for τ 1. Hence, starting with period t +1, the optimal investment policy cycles through investments I t 3, I t 2, I t 1, I t, I t 3, and so forth. Under this policy, the firm s value is completely determined by the history of its latest four investments. V t = (pq max I t 3 ) γ +(pq max I t 2 ) γ = = 1 r pq max γ + γ I t 3... γ 4 + γ I t = 1 r pq max γ 1 γ 4 I t 3 γ2 1 γ 4 I t 2 γ3 1 γ 4 I t 1 γ4 1 γ 4 I t. (1) 2.1 Straight-Line Depreciation Assume that the company prepares financial statements in accordance with the straight-line depreciation rule. Thus, assets are capitalized in the acquisition period and then depreciated evenly over the next four periods. At the end of period t, the accounting system reports the following information: revenues, R t, R t = p (I t 1 + I t 2 + I t 3 + I t 4 ), depreciation, D t, D t = 1 4 I t I t I t I t 4, 7

8 book value at date t, BV t, BV t = I t I t I t I t 3, and cash flows to investments, I t. Given clean surplus accounting, investors can also infer the beginning of period book value of assets, BV t 1,as BV t 1 = BV t + D t I t. Define the firm s state at date t as the history of its past five investments, θ t =(I t,i t 1,I t 2,I t 3,I t 4 ) 0. Note that all accounting numbers above are linear combinations of the latest five investments and therefore are completely determined by the state vector θ t. The main idea of the example is to demonstrate that, depending on the choice of accounting rules, current period financial statements may be sufficient or insufficient for valuation purposes. While straight-line depreciation may seem to be a natural method to account for assets whose productivity is constant over time, it turns out that this rule entails a loss of value-relevant information. To demonstrate this claim, it suffices to describe two hypothetical firms, operating in the same market, and generating identical financial statements, yet because of different investment histories their intrinsic values differ. Observing only the financial statements, investors will not be able to figure out the underlying investment history, and, consequently, they will not be able to value the two firms correctly. Specifically, assume that q max =40and consider the following two histories at date t: θ (1) t =(10, 10, 10, 10, 10) and θ (2) t =(10, 15, 0, 15, 10). Firm 1 is in its steady state and invests 10 dollars in every period. Firm 2 cycles through investments 15, 0, 15, 10, also implementing capacity of 40 in every period. At date t, these two histories lead to exactly the same financial statements. Indeed, for both firms, we have R t =40p, D t =10, BV t =25, I t =10. On the other hand, applying equation (1) to these histories, one can compute the valuations 8

9 of the two firms at date t: t = 1 r pq γ γ2 max γ4 1 γ I 4 t 2 10 γ3 γ γ4 1 γ I 4 t V (1) and t = 1 r pq γ γ2 max γ4 1 γ I 4 t γ4 1 γ I t. 4 V (2) Since γ 3 < (γ 2 + γ 4 ) /2, thevalueoffirm 1 is greater than that of firm 2: V (1) t >V (2) t. However, observing only the latest financial statements, it is impossible to infer which of the two firms generated them. Hence, the valuation problem is not solvable in this case, and straight-line depreciation is informationally insufficient for assets corresponding to the one-hoss shay pattern. To further illustrate this insufficiency result, consider the application of the residual income valuation model to firms 1 and 2. Residual income in period t, RI t,isdefined as the difference between revenues and the aggregate historical cost of capacity, H t,thelatter being the sum of depreciation and an imputed charge on the beginning of period book value: RI t = R t H t R t D t rbv t 1. It is well known that regardless of the accounting rules, value can be expressed by means of the residual income valuation formula: 5 V t = BV t + X γ τ RI t+τ. τ=1 In our example, book values of both firms are equal to 25 at date t 1. Given straight-line depreciation, residual income of firm 1 is constant over time and always equal to RI (1) t+τ =40p 10 25r. One can check that firm s 2 residual income in period t is also equal to 40p 10 25r, but from period t +1onwards, it will iterate through the following four values 40p 10 25r, 40p 10 30r, 40p 10 20r, 40p 10 25r,... 5 See, for instance, Penman (2007). 9

10 Therefore, the future residual income processes of the two firms are different, although they operate in equivalent environments and report the same accounting numbers at date t. For this reason, the current financial statements do not convey enough information to predict future residual earnings, even though investors may correctly anticipate future conditions in the product market. 2.2 Annuity Depreciation We now consider an alternative accounting policy - the annuity depreciation rule. Assets are again capitalized at historical cost in the acquisition period and then fully depreciated over their useful life, yet the depreciation charges now compound at the rate of 1+r: d τ+1 =(1+r) d τ, (2) where d τ is the depreciation charge per one dollar of assets in period τ. Since assets are fully depreciated, the depreciation charges must satisfy the relation: d 1 + d 2 + d 3 + d 4 =1. (3) Equations (2) and (3) imply that: d τ = rγ5 τ (4) 1 γ 4 for 1 τ 4. One can also check that the book value at date t of one dollar of assets acquired in period t τ is given by bv τ =1 d 1... d τ = d τ d 4 = 1 γ4 τ 1 γ 4 for 0 τ<4. Therefore, the aggregate book value and depreciation at date t are: D t = BV t = I t + 1 γ3 1 γ 4 I t γ2 1 γ 4 I t γ 1 γ 4 I t 3, (5) rγ4 1 γ I 4 t 1 + rγ3 1 γ I 4 t 2 + rγ2 1 γ I 4 t 3 + rγ 1 γ I t 4. (6) 4 10

11 It should also be noted that for any investment history the aggregate historical cost of capacity in a given period is proportional to capacity utilized in that period: H t+τ = D t+τ + rbv t+τ 1 = r 1 γ (I 4 t+τ I t+τ 4 ) = r 1 γ K t+τ. 4 (7) In later sections, this observation will be shown to be a special case of a broader replacement cost accounting property for assets conforming to the one-hoss shay pattern. To demonstrate this link, let me also describe the annuity rule by invoking the concept of a hypothetical perfectly competitive rental market for capital assets. If a perfect rental market were to exist, then the rental price of a unit of capacity, c, would be such that a rental firm would exactly break even over time. If the rental firm invests one dollar in period t and then rents out the resulting capacity in the following four periods, its net present value is: 1+γc + γ 2 c + γ 3 c + γ 4 c. Since the rental market is assumed to be perfectly competitive, the expression above must be equal to zero. Therefore, the competitive rental price of a unit of capacity is given by: c = 1 γ + γ 2 + γ 3 + γ 4 = r 1 γ 4. (8) Under replacement cost accounting, book value at date t of an asset purchased in period t τ is defined as the fair value of that asset in the hypothetical rental market at date t. For τ<4, a unit investment in period t τ adds one unit of capacity in periods t+1,...,t+4 τ. To replace this stream of capacity, the firm would need to incur the following cost: γc γ 4 τ c = 1 γ4 τ 1 γ 4 = bv τ. Therefore, the annuity rule corresponds to replacement cost accounting if a perfect rental market exists. Also, combining equations (7) and (8), one obtains H t+τ = ck t+τ. (9) Thus, the annuity depreciation rule also implies that the aggregate historical cost of capacity is equal to the replacement cost of capacity utilized in the current period. 11

12 To show that the annuity rule is informationally sufficient, one can apply the residual income valuation approach. Property (9) implies that on the optimal path of future capacity levels, residual income is given by: RI t+τ = R t+τ H t+τ = pk t+τ ck t+τ =(p c) q max (10) for τ 0. Therefore, the firm s value can be computed as follows X V t = BV t + γ τ RI t+τ = BV t + 1 r RI t. (11) τ=1 Beginning-of-period book value can be recovered from the financial statements at date t in the following way: BV t 1 = BV t + D t I t. Substituting this expression into (11), one obtains: 6 Observation 1 Given annuity depreciation, V t = BV t + 1 r (R t D t rbv t 1 ) = 1 r R t 1 rγ D t + I t. V t = 1 r R t 1 rγ D t + I t. Since the firm s value can now be expressed in terms of the current accounting data, we have demonstrated the informational sufficiency of the annuity depreciation rule. Central to this result is the fact that under the annuity rule residual income is constant over time and equals (p c) q max for any optimized investment history. It will be shown below that (p c) q max can be viewed astheeconomicprofit ofthefirm in all periods. Since it was assumed that both capital and output markets are stationary, the fact that optimized economic profits are also stationary should not come as a surprise. In this simplified environment, I showed that under annuity depreciation the current residual income provides sufficient information for predicting future 6 in this example, the book value of assets drops out from the valuation formula because the product market is assumed to be stationary. The general model allows for growth in the product market, and then the coefficient on the book value in the valuation formula will be different from zero. 12

13 residual incomes. This contrasts with the findings in the straight-line scenario, where current accounting data was insufficient for predicting future residual earnings. To conclude the discussion of this example, it is instructive to compare replacement cost accounting with fair value accounting. When the output price is high enough to cover the marginal costs of production, specifically when p is greater than c, bookvaluesunderthe annuity rule will be less than the present values of future cash flows, since residual earnings will be always positive by equation (10). Therefore, the annuity rule is more conservative than fair value accounting in the sense that accounting book values are always below market values. 3 Model Description 3.1 Transactions Consider a firm that invests in a single type of long-lived assets and produces a single output good. I will assume that the cash cost of one unit of physical assets is constant over time and equal to k. 7 Let I t be the capital expenditure in period t, startingatdatet 1 and ending at date t. Then, the number of asset units acquired in that period is I t /k. Eachunit of the physical asset is idle in the acquisition period and then generates capacity to produce x 1,...,x T units of output in the T periods of its useful life, where x 1 x 2... x T > 0 and T 2. For notational convenience, I will also define x 0 =0to be the asset s productivity in the acquisition period. Hence, the total capacity available in period t, K t,isgivenby: K t = k 1 (I t x 0 + I t 1 x I t T x T )=k 1 x θ t, where x (x 0,..., x T ) 0 will be referred to as the asset s productivity profile and θ t (I t,..., I t T ) 0 is the relevant investment history at date t. If all x τ are equal, I will call the productivity profile the one-hoss shay pattern. In period t, thefirm faces an inverse demand curve, P t (q t ),whichdefines price as a function of quantity of output sold, q t.letr t (q t ) denote the corresponding revenue function: R t (q t )=P t (q t ) q t. In the interest of parsimony, I assume that the firm is all-equity financed, there are no oper- 7 For now, I present the model as one of certainty about the conditions that the firm experiences in the future in its capital asset and product markets. Section 6 below discusses several ways to introduce uncertainty in the model. 13

14 ating expenses and taxes, and all free cash flows are disbursed to shareholders immediately. Investors are interested in valuing the company under the assumption that managers have perfect information and always act in the best interests of the firm s owners. An investor s valuation of the company at date t is equal to the present value of future cash flows: V t = max I t+τ,τ=1,... X γ τ (R t+τ (q t+τ ) I t+τ ), (12) τ=1 where γ = 1,ris the firm s cost of capital, and I 1+r t+τ is chosen optimally in every period. Arrow (1964) showed that under certain conditions the optimization problem in (12) is time-separable. In particular, this separability will hold if the following no-excess capacity condition is met. Assumption 1 (No-Excess Capacity Condition) Investors assume that demand curves shift out over time, in the sense that P t+1 (q) P t (q) for any q, t. Under this assumption, the optimal capacity levels are nondecreasing over time, while the capacity from assets already in place at any given date is nonincreasing. Therefore, the firm never ends up with excess capacity. The firm s value can be then rewritten as: V t = max I t+τ,τ=1,... X γ τ (R t+τ (K t+τ ) I t+τ ). (13) τ=1 Define the marginal cost of capacity in period t + τ, c t+τ, as the incremental cost to the firm, in t + τ-period dollars, of generating one additional unit of capacity in that period, holding all other capacity levels fixed. Since the productivity profile and the price of assets are invariant over time, one might expect the marginal cost of capacity to be also constant. Consequently, the subscript t + τ will be omitted. Arrow (1964) showed that the marginal cost is given by: k c = x 1 γ x T γ. (14) T The intuition behind this result can be demonstrated by considering again the notion of a hypothetical rental market for capital assets and by assuming that this hypothetical market is also perfectly competitive. As in the numerical example in Section 2, c can be shown to 14

15 be equal to the rental price per unit of capacity at which a competitive rental firm exactly breaks even over time. Indeed, assume that the rental firm invests one dollar in period t and rents out the resulting capacity in periods t +1 through t + T. Then, present value of its cash flows is 1+γ x 1 k c + x γ2 2 k c x γt T k c. The value of c in expression (14) equates the present value of rental firm s revenues with the costs to generate those revenues. Since the rental business is competitive, the producing firm is indifferent between investing in assets at a unit price k and renting capacity at a unit price c. Hence, by internalizing a cost of c per unit of capacity, the producing firm will generate the first-best investment policy in acquiring new capacity. On the optimal path, each K t+τ must be chosen so that the marginal revenue from an additional unit of capacity is equal to the marginal cost of that unit, c: Rt+τ 0 (K t+τ )=c. With an implicit reference to the notion of a hypothetical rental market, I will call ck t+τ the replacementcostofcapacityin period t + τ. Sincec is the marginal cost of capacity, it is also natural to label ck t+τ the economic cost, andthedifference between revenues and ck t+τ -theeconomic profit of the firm: π t+τ = R t+τ (K t+τ ) ck t+τ. ArelatedconceptthatIwillusethroughoutthepaperisthereplacementcostofassets in place at date t. As was demonstrated in the numerical example, two firms operating in the same economic environment and implementing equal capacity in every period, can have different valuations. This difference reflects that the composition of assets in place differs for the two firms at a given date. Clearly, if two firms utilize the same amount of capacity in period t, butfirm s 1 capacity is newer than that of firm 2, then firm1willhaveagreater intrinsic value. Intuitively, assets in place can be essentially viewed as cost-savings in future periods. To quantify these savings, let v τ denote the replacement cost factors at date t of a dollar of investment made in period t τ, defined as the present value of hypothetical rental payments that the firm would need to incur in order to replace this investment moving forward. Since the investment was made τ periodsago, itwillprovidethefollowingstream of capacity starting with period t +1 1 k x τ+1,..., 1 k x T. 15

16 Therefore, v τ = γc 1 k x τ+1 + γ 2 c 1 k x τ γ T τ c 1 k x T. Substituting for c, the replacement cost factor v τ can be simplified to v τ = x τ+1γ x T γ T τ x 1 γ x T γ T. (15) Note that v 0 =1, indicating that the replacement cost of the investment just made is its historical cost. The aggregate replacement cost of assets in place at date t is the sum of replacement costs of assets still productive at that date: RC t = v 0 I t v T I t T. 3.2 Financial Reporting The accounting system aggregates the information about past transactions of the firm into the financial statements. Depreciation is the only accrual considered in the model. Therefore, investors learn the following four numbers from the income statement and the balance sheet at date t: revenues for the latest period, R t, depreciation, D t,netincome,inc t = R t D t, and book value of assets at date t, BV t. Investors further learn the latest investment, I t,from the statement of cash flows. I will assume that depreciation is computed according to a fixed schedule d =(d 0,..., d T ) 0. This schedule can be tailored to the anticipated physical decay of assets in the sense that d can depend on the productivity profile x. Total depreciation in period t then becomes: D t = d 0 I t d T I t T = d θ t. Let bv τ denote the share of investment I t that remains capitalized at the end of period t + τ, and let bv =(bv 0,..., bv T ) 0. I will restrict attention to depreciation rules satisfying the usual "tidiness" requirement that: τx bv τ =1 d i, i=0 and TX d i =1. i=0 Hence, bv T =0. Thus, the book value of each investment changes only due to depreciation charges related to that investment, and assets are fully depreciated over their useful life. In 16

17 this notation, the aggregate book value at date t is equal to: BV t = bv 0 I t bv T I t T = bv θ t. To summarize, investors observe the following information set at date t: 8 I t = {R t,d t,bv t,i t }. Note that book value, depreciation, and the latest investment are dot products of the state vector θ t with vectors bv, d, andi (1, 0,..., 0) 0, respectively. Also, revenues are a function of capacity in period t, which is proportional to the dot product of θ t and x: I t = R t k 1 θ t x, d θ t, bv θ t, i θ t ª. While the investors information set does not contain the state vector θ t itself, investors do observe a number of linear transformations of this vector. The linear aggregation structure is crucial for the informational sufficiency results discussed below. It will be convenient to define the aggregate historical cost of capacity in period t, H t,as the sum of depreciation expense and a cost of capital charge on the beginning book value of assets: H t = D t + rbv t 1. The historical cost is a function of the state vector at date t, sincebothd t and BV t 1 are determined by that vector: H t = d 0 I t +(d 1 + rbv 0 ) I t (d T + rbv T 1 ) I t T. Let z 0 = d 0 and z τ = d τ + rbv τ 1 for 0 <τ T and let z =(z 0,..., z T ) 0 be the vector of historical cost charges. Then, H t = z θ t. Prior literature has established that there exists a one-to-one mapping between depreciation charges and historical cost charges (see e.g. Rogerson, 1997, and Reichelstein, 1997). In particular, it can be shown that the depreciation vector will satisfy the clean surplus 8 Net income is contained in I t, but it has no incremental information content beyond revenues and depreciation. Also, I will routinely assume that BV t 1 is in I t,sincebv t 1 = BV t + D t I t. 17

18 condition if and only if the corresponding z-vector satisfies: TX z τ γ τ =1. (16) τ=0 Following Rogerson (2008b), I now define replacement cost accounting, or the replacement cost rule, in terms of its corresponding z-vector and then check that condition (16) is satisfied. Let zτ be equal to the replacement cost of capacity provided in period t by a unit investment made in period t τ : zτ = k 1 x τ cx τ =. γx γ T x T Intuitively, this rule allocates historical cost charges to a particular period, τ, inproportion to the capacity that the asset generates in that period. Condition (16) is satisfied for the vector z : TX zτγ τ 1 TX = x (γx γ T τ γ τ =1. x T ) τ=0 Given this rule, the aggregate historical cost in period t is indeed equal to the replacement cost of capacity, irrespective of the investment history: τ=0 H t = z θ t = k 1 cx θ t = ck t. (17) Finally, residual income in period t is the difference between revenues and the historical cost of capacity: RI t = R t D t rbv t 1. (18) Let d, bv denote the depreciation and book value schedules corresponding to z and let Dt, BVt be the aggregate depreciation and book values under this rule. By (17), residual income under the replacement cost rule, is equal to the firm s economic profits RI t = R t H t = R t ck t (19) for any t. The following benchmark result is an application of the residual income valuation formula for the special case of replacement cost accounting. 9 Proposition 1 The value of the firm at date t is equal to the sum of the replacement cost 9 Related results were established in the economics and finance literature, see e.g. Thomadakis (1976) and Lindenberg and Ross (1981). However, in these studies capital assets are usually assumed to be infinitelylived and their productivity is assumed to decline geometrically over time. For the purposes of this paper, it is important to state the result in Proposition 1 for general productivity patterns. 18

19 of assets in place and the present value of future optimized economic profits: V t = v θ t + X τ=1 max (γ τ (R t+τ (K t+τ ) ck t+τ )). (20) K t+τ Given replacement cost accounting, the first component of V t corresponds to the book value of assets, and the second component is the present value of future residual earnings. The proof of Proposition 1 demonstrates that the book values corresponding to the z - rule are equal to the replacement cost of assets in place, that is: BV t = RC t, hence the term replacement cost accounting. The claim in Proposition 1 then follows from the residual income valuation formula. Note also that there is a clear distinction of economic stocks and flows in the present model: the replacement cost of assets in place can be viewed as the value of the firm s stock at date t, while future economic profits are the flow variables. If the firm operates in a competitive environment, then future economic profits are zero and the value of the firm is equal to the replacement cost of assets in place. In much of the discussion above, the replacement cost rule was described with a reference to the rental market for capital assets. It is important to note that under the no-excess capacity condition (Assumption 1), the existence of such a market is not required. When the output market expands, the firm seeks to increase its capacity in every period. Under this assumption, the firm will never find it desirable to rent out its capacity. Therefore, the absence of a rental market does not affect the firm s first-best investment policy nor does it affect the firm s valuation. Replacement cost accounting may then be viewed as a particular depreciation rule corresponding to historical cost accounting. 4 Informationally Sufficient Accounting 4.1 Sufficiency of Replacement Cost Accounting In valuing the company, investors seek to estimate (i) the replacement cost of assets in place and (ii) the present value of future optimized economic profits. Estimating future profits naturally requires an assessment of how the output market conditions evolve over time. In this regard, the following assumption will be imposed on the evolution of the firm s inverse demand functions A similar assumption is invoked in Nezlobin, Rajan, and Reichelstein (2008). 19

20 Assumption 2 (Proportionate Growth Assumption) Market demand evolves such that P t+τ (g t+τ q)=p t (q) (21) for all q, whereg =(g t+1,..., g t+τ,...) and g t+τ is the cumulative output market growth factor between period t and period t + τ. 11 Assumption 2 states that for all price points, demand increases by a factor of g t+τ from period t to period t+τ. The no-excess capacity condition is implied when all g t+τ are greater than one and are non-increasing: 1 g t+1... g t+τ... As a consequence, the output market is non-declining, and since the capacity from assets in place is non-increasing, the firm never expects to end up with excess capacity. The key property of the proportionate growth parametrization is that the future optimal prices and capacity levels can be expressed as simple functions of the current optimal capacity and price. To see this, observe that if P t (K t ) K t ck t is maximized at some capacity level, Kt,then µ Kt+τ Kt+τ P t+τ (K t+τ ) K t+τ ck t+τ = g t+τ P t g t+τ g t+τ c K t+τ g t+τ will be maximized at Kt+τ = Kt g t+τ. Therefore, optimal capacity levels will follow the growth pattern g, optimal prices will be constant, and revenues and economic profits will also grow according to g. It should also be noted that to make such projections, investors do not need to know the structural form of the inverse demand curves. I now specify in more detail the information available to different parties. At date t, the manager has perfect information about past transactions of the firm and knows the inverse demand function in period t +1. This information allows the manager to implement the path of the first-best investments. The accounting system tracks all transactions of the firm 11 If µ t+τ is the market growth rate from period t + τ 1 to t + τ, then g t+τ = τy 1+µt+i i=1 20

21 and takes into consideration the asset s productivity profile, x. This knowledge can be used in choosing an appropriate depreciation rule for the firm s assets. Investors face the problem of estimating V t, on the basis of only the aggregate financial statements. More specifically, it will be assumed that investors do not observe any of the investments in θ t,exceptforthe latest one, neither do they know the acquisition cost of capital assets, k. Also, while they are not knowledgeable of the exact shape of the demand curve, they assume that demand will grow proportionately at all price levels, according to the pattern g. I allow for the possibility of investors observing output prices and the productivity profile of assets. In particular, the informational structure described above precludes investors from inferring capacity costs from revenues only. For example, if investors knew k, they would be able to infer the marginal cost of capacity using equation (14) and, then, compute the aggregate economic cost by dividing revenues by the output price and multiplying the resulting capacity by c. Alternatively, if they knew the exact shape of the demand curve, they could infer c asthemarginalrevenueattheoptimalpoint. Sincemyinterestisinmodelingthe accounting system as the primary vehicle of information transfer to investors, both of these inferences are rendered infeasible. Finally, I assume that future growth factor projections, g, cannotbeincorporatedinto the financial statements, either because these projections are not considered verifiable to the accounting system, or because these projections may be investor-specific andtheac- counting system needs to accommodate all heterogeneous investors. 12 A consequence of this assumption is that fair value accounting, under which BV t is always equal to V t, is rendered infeasible, because V t inherently depends on g. Accounting rules will be called informationally sufficient if there exists a value estimate which uses as inputs only the information available to investors at date t, andwhichcaptures the value correctly for any projection of growth in the product market. If the accounting rules used for financial reporting do not meet this criterion, then the valuation problem is generally not solvable. Definition 1 A depreciation rule, d, issaidtobeinformationally sufficient if there exists afunction ˆV t (I t, g) such that V t (θ t,p t ( ),k,g) = ˆV t (I t, g) for any market demand P t ( ), andanyvector(θ t, g,k). 12 The current model can be extended without much additional machinery to accomodate certain forms of uncertainty. Examples of possible extensions are discussed in Section 6 below. 21

22 Given the result in Proposition 1, replacement cost accounting will be informationally sufficient provided investors can estimate future residual earnings from the current financial statements. This forecast is possible if future market demand conforms to the proportionate growth assumption. Proposition 2 Replacement cost accounting is informationally sufficient. The intrinsic value of the firm is given by V t = BVt + αrit, (22) where α = P τ=1 g t+τγ τ. Proposition 2 identifies informationally sufficient systems for different productivity patterns. As mentioned above, the replacement cost rule for one-hoss shay productivity is the annuity depreciation, therefore the annuity rule is an informationally sufficient system for assets with this productivity pattern. This observation was the basis of the numerical example discussed in Section 2. Proposition 2 suggests two value-relevant dimensions of the investment history. 13 Since K t = k 1 x θ t, it follows that V t = v θ t + α R t k 1 x θ t k 1 x θ t. (23) Thus the value of the firm depends on two dot products of the investment history with the vectors - v and x. If we consider another firm, operating in the same environment, with investment history θ (1) t,whichdiffers from θ t in a dimension orthogonal to both v and x, then the value of that firm will be exactly the same. On the other hand, variations in the investment history along any of these two dimensions will lead to different valuations. Therefore, it is natural to label v and x the value-relevant dimensions of θ t. Since under replacement cost accounting book values correspond to replacement cost of assets in place, bv = v, and the vector of historical charges, z,isproportionaltox, I will also refer to bv and z as the value-relevant dimensions. Intuitively, the first dimension conveys information about the replacement cost of assets in place at date t, whereas the second dimension defines revenues, replacement cost of capacity, and the economic profit inperiodt. Relative weights of these two dimensions depend on α, which in turn, is a function of the growth rates in sales revenues. For a competitive firm, economic profits are zero and v is the only relevant 13 Alternative notions of value-relevant information are discussed in Holthausen and Watts (2001) and Barth et. al. (2001). In Section 5, I discuss in more detail the connection between the notion of value relevance in Barth et. al. (2001) and the one that I introduce in this paper. 22

23 dimension. If the firm earns some economic rents in the future, then, as α increases, a relatively larger share of value is due to future economic profits, and differences in histories along vector x lead to greater differences in valuations. 4.2 A Uniqueness Result I now turn to the issue of characterizing the class of informationally sufficient accounting rules. Clearly there is one degree of freedom associated with replacement cost accounting that does not impede its informational properties. This degree of freedom corresponds to the possibility of partial direct expensing (or write-ups) in the acquisition period. Consider the following class of schedules: bv =(1 λ) bv, (24) where λ<1 is some constant. It is readily verified that all accounting rules in this class are informationally sufficient. Recall that d 0 =0and bv0 = v 0 =1. Therefore, bv 0 =(1 λ). Depreciation schedules defined by (24) have the property that in the acquisition period a share λ of the investment is directly expensed (or written-up), d 0 = λ. In future periods, the amount initially capitalized is depreciated in proportion to the replacement cost rule: d τ = bv τ 1 bv τ =(1 λ) bv τ 1 (1 λ) bv τ =(1 λ) d τ. Book values under these schedules are proportional to the replacement cost book values at all times: BV t =(1 λ) BVt. (25) The parameter λ can be viewed as a degree of unconditional conservatism of the accounting system in use. 14 Positive values of λ define rules more conservative than the replacement cost rule, while negative values of this parameter define more liberal rules. The one-dimensional family of replacement cost accounting rules corresponding to λ<1 will be called generalized 14 In the empirical part of their paper, Rajan, Reichelstein, and Soliman (2008) operationalize conservative accounting by the proportion of new investments that are expensed directly, e.g. R&D and advertising, relative to total investments. 23